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The Idea of Holonomy

Unless you’ve studied differential geometry, you have probably never heard of holonomy; a quick search online turns up dense paragraphs about “smooth manifolds” and “parallel transport.” Yet at a recent MAA Distinguished Lecture, Robert Bryant, director of the Mathematical Sciences Research Institute, was able to explain this seemingly obscure concept with the help of a basketball.

In fact, whenever Bryant gives a nontechnical introduction to holonomy—a concept he regards as simultaneously fundamental and underpublicized—he makes sure to have one of those bouncy orange spheres on hand.

"I was at Duke for 20 years," Bryant told listeners at the MAA Carriage House on October 14, “and that may have influenced a little bit my choice of props.”

Holonomy, it turns out, is a tool for understanding constrained motion, situations in which you ultimately want to move in more dimensions than you have freedom to move in at any instant.

Robert Bryant uses a basketball to present "The Idea of Holonomy."

Think of a handcart. (Bryant couldn’t bring such a bulky item on the plane with him, but the MAA obligingly provided one for demonstration purposes.) While the cart can move only backward or forward at any particular time, it should, for maximum usefulness, be navigable around an entire floor.

A basketball invites another example: a ball rolling on a table without slipping or twisting. “Can I roll the ball from any point to any other point and have it wind up in a given orientation that we want?” Bryant asked. The answer, he said, has consequences for fields from robotics to control theory.

It’s helpful, though, to first think about holonomy effects in a simple case, in which a three-dimensional object doesn’t roll but instead tumbles, touching the surface with one distinct face after another.

Take the cubical analog of the basketball question: Can you start with a cube on any square of a grid and tumble it to any other square such that it arrives in a particular one of its 24 possible orientations?

With the aid of a grid chalked on the blackboard, a cube constructed of brightly colored Polydrons, and occasional excursions into group theory, Bryant reduced an initially overwhelming assortment of paths to a manageable number. In the end, the problem hinged on how the orientation of a cube changes when it is tumbled around a particular vertex .

Standing at the board, Bryant tumbled the cube and had the audience help keep track of the red, blue, green, and yellow faces. The verdict? Movement on the grid is restricted: You can only ever get 12 of the 24 orientations. “The holonomy is the alternating group,” Bryant said.

Robert Bryant presents "The Idea of Holonomy" as part of MAA's Distinguished Lecture Series

After honing his audience’s holonomic instincts with several more examples—Bryant produced from his grocery bag a tetrahedron, an octahedron, and finally, an icosahedron—he returned to the basketball.

As with the cube, Bryant boiled the problem down to a simple question: Is the sort of rolling we’re talking about—no slipping or twisting—rotation about a fixed axis? If not, Bryant argued, then we can indeed arrive at any point in any orientation we please. So what are you waiting for? Grab a basketball and investigate!

In the real world, of course, planes aren’t perfectly flat and balls aren’t completely spherical. Things are not, in other words, as tidy or well behaved as Bryant’s cart, balls, and Polydron creations. But there’s theory, Bryant said, “that allows us to do calculus, to not only determine whether you can move freely in a holonomically constrained space but also how you can move most efficiently.”

Sophisticated mathematics allows Bryant to tackle thorny real-world problems, but he appreciates the humble basketball too. Although he conducts much of his work using only calculation, Bryant knows that props help others grasp concepts needed to introduce a wider audience to holonomy. —Katharine Merow